There is no specific day and year stated in the first scene at Daisy's house, although, there are clues. The whole book takes place in the summer of 1922 in New York City. While at the house, Daisy states
"‘In two weeks it’ll be the longest day in the year.’ She looked at us all radiantly. ‘Do you always watch for the longest day of the year and then miss it? I always watch for the longest day in the year and then miss it.’ " -Daisy, The Great Gatsby
The longest day of the year is the summer solstice, which is usually June 21st, the first day of summer, So exactly two weeks before is June 7. So when they were first at the house, it was around June 7th, 1922.